GATE 2026 EE Q.40: solution and answer

GATE 2026 · EEQ.40 · 2 marks · MCQ

The operating characteristic of a reactance relay is given by X ≤ 1 Ω, where X is the reactance calculated by the relay. Its operating characteristic in the admittance plane (G-B plane, where G and B denote conductance and susceptance, respectively, expressed in mho) is given by:

(A)
G^2 + (B + 0.5)^2 ≥ 1/4
(B)
B ≥ 1
(C)
(G − 1)^2 + B^2 ≤ 1/2
(D)
G^2 + (B − 1)^2 ≥ 1/2

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Answer and solution

ANSWER

Option (A): $G^{2} + (B + 0.5)^{2} \geq \frac{1}{4}$ .

Official answer: ASource: Official answer key, GATE 2026 EE (IIT Guwahati)

Let the relay see impedance $Z = R + j X$ and admittance $Y = \frac{1}{Z} = G + j B$ .
Rationalise: $Y = \frac{R - j X}{R ^{2} + X ^{2}}$ , so $G = \frac{R}{R ^{2} + X ^{2}}$ and $B = - \frac{X}{R ^{2} + X ^{2}}$ .
Then $G^{2} + B^{2} = \frac{R ^{2} + X ^{2}}{( R ^{2} + X ^{2} ) ^{2}} = \frac{1}{R ^{2} + X ^{2}}$ , so $X = - \frac{B}{R ^{2} + X ^{2}} = - \frac{B}{G ^{2} + B ^{2}}$ .
The operating condition $X \leq 1$ becomes $- \frac{B}{G ^{2} + B ^{2}} \leq 1$ . Since $G^{2} + B^{2} > 0$ , multiplying through does not flip the inequality: $- B \leq G^{2} + B^{2}$ .
Rearrange: $G^{2} + B^{2} + B \geq 0$ .
Complete the square: $G^{2} + (B + \frac{1}{2})^{2} \geq \frac{1}{4}$ .
This is the region outside a circle of radius $0.5 mho$ centred at $(G, B) = (0, - 0.5)$ , passing through the origin. It matches option (A).

KEY IDEA

Write the admittance as $Y = 1/ Z$ . Then $X = - B / (G^{2} + B^{2})$ , and a line $X = const$ in the impedance plane becomes a circle through the origin in the admittance plane. The inequality $X \leq 1$ maps to the outside of the circle $G^{2} + (B + 0.5)^{2} = 0.25$ .

WATCH OUT FOR

Forgetting the minus sign in $B = - X / (R^{2} + X^{2})$ , which puts the circle centre at $+ 0.5$ and leads to a wrong option. Another slip is reversing the inequality. Both $X$ and $B$ carry signs, but the multiplier $G^{2} + B^{2}$ is positive, so the direction stays the same and the region is outside the circle.

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GATE 2026 EE Q.40: solution and answer

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